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A086792
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Orders of finite groups G with the property that the sum of the orders of all the proper normal subgroups of G equals the order of G.
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1
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OFFSET
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1,1
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COMMENTS
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The only Abelian groups with this property are the cyclic groups C_n where n is a perfect number, so this sequence can be seen as a groups analogy of perfect numbers.
Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) computed the orders of the non-Abelian groups in the sequence up to n=500 and commented "In general, if 2^n - 1 is a Mersenne prime, then 2^(n-1)*(2^n - 1) is a perfect number and the group with presentation < x,y | x^(2^n-1) = 1, y^(2^n) = 1, y^-1 x y = x^-1 > has order equal to the sum of the orders of its proper normal subgroups." So if n is an even perfect number, 2n also belongs to this sequence (the numbers 12 and 56 above).
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LINKS
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Table of n, a(n) for n=1..9.
Tom Leinster, Perfect numbers and groups
Sci.math, Source for sequence
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CROSSREFS
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Cf. A000396.
Sequence in context: A036833 A009242 A032647 * A064987 A057341 A068412
Adjacent sequences: A086789 A086790 A086791 * A086793 A086794 A086795
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KEYWORD
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nonn
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 04 2003
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STATUS
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approved
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